A Numerical Algorithm for Block-diagonal Decomposition of Matrix ∗-algebras

Motivated by recent interest in group-symmetry in the area of semidefinite programming, we propose a numerical method for finding a finest simultaneous block-diagonalization of a finite number of symmetric matrices, or equivalently the irreducible decomposition of the matrix ∗-algebra generated by symmetric matrices. The method does not require any algebraic structure to be known in advance, whereas its validity relies on matrix ∗-algebra theory. The method is composed of numerical-linear algebraic computations such as eigenvalue computation, and automatically makes the full use of the underlying algebraic structure, which is often an outcome of physical or geometrical symmetry, sparsity, and structural or numerical degeneracy in the given matrices. Numerical examples of truss and frame designs are also presented.